morphism group of Q, then every extension of Q by H which is associated with 4, is a composite of any one of them, singled out arbitrarily, with an extension of C by H which is associated with the induced homomorphism of H into the group of automorphisms of C. The proof given by Eilenberg and MacLane makes essential use of factor sets and is therefore not applicable to topological group extensions. In ?1, we shall give a different proof which is applicable to general topological groups. However, in the general case, the theorem requires an additional topological restriction (see Defn. 1.1). Because of the necessity of verifying that our various constructions do not compel us to leave the category of Lie groups, we shall word our proof for Lie groups. In the rest of this paper, we shall confine our attention to extensions of abelian Lie groups. In particular, if C is an abelian Lie group, and if C1 denotes the connected component of the identity in C, we shall study the relationships between the extensions, by a fixed Lie group H, of C1, C, and C/Cl . For the main results, we assume that H is connected. In that case, if 4,t is a homomorphism of II into the group A(C) of the continuous open automorphisms of C which is determined by some extension of C by H the automorphisms belonging to Av(H) map every component of C onto itself, and the induced homomorphism,